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Deuteron-deuteron scattering is the simplest example of a collision between two composite nuclear systems. Its experimental study provides a test of features predicted by multiple nuclear scattering theories that can be made only when both projectile and target are composite. There are several theoretical approaches to this problem 1 " 5 and we compare our data with some of these that have been applied specifically to elastic dd scattering. 6 " 8 Previous studies have been made of particle-deuteron collisions (pd,pd,ird, Kd) where the Glauber multiple-scattering theory, 2 with a recent extension which takes into account the quadrupole deformation of the deuteron, 9 has proved quite successful. 10,11 We find that the Glauber theory is also qualitatively correct in interpreting dd elastic scattering, but more detailed theoretical calculations must be done before quantitative success can be claimed. Such studies, applied to composite systems of familiar particles, may help in formulating theoretical models which treat hadron-hadron scattering as collisions between composite systems of unknown subparticles. 12The elastic and total cross sections for deuteron-deuteron interactions at 680, 920, 1180, 1500, 1750, and 2120 MeV/c are presented. The data are compared with theoretical calculations and shown to be in qualitative agreement with predictions based on the Glauber multiple-scattering formalism. In particular the existence of large-angle deuteron-deuteron elastic scattering provides evidence for the "simultaneous" scattering amplitudes predicted by the Glauber theory.